Set up your grill for direct grilling and preheat to medium. Using a sharp knife, cut the top third off each artichoke and rub with lemon to prevent discoloring. Using scissors, cut the spines off the artichoke leaves and rub with the lemon. Trim off the ends of the stems, but leave as much of the stem intact as possible. Using a grapefruit spoon or melon baller, scrape out the hairy chokes. Squeeze lemon juice into the cavities. Generously brush the artichokes inside and out with oil and season with salt and pepper.

Place the artichokes on the grill, stem ends up, and grill until the cut side is nicely browned, about 30 minutes. Rotate the artichokes from time to time to prevent them from burning.

Invert the artichokes (the stems should fit through the bars of the grateif not, simply fold to one side), brush generously with oil, and place the minced garlic and the chopped mint and parsley in the cavity of each. Season again with salt and pepper. Cover and continue grilling the artichokes, basting with oil, until very tender, 30 to 45 minutes. You'll need 1 to 1 1/4 hours grilling in all. When the artichokes are done, the leaves should pull away easily. Drizzle with any remaining oil and serve. Provide finger bowls and napkins.